Supercond. Sci. Technol. 10 (1997) 523–542. Printed in the UK
PII: S0953-2048(97)78196-X
TOPICAL REVIEW
Characterization of high-temperature
superconductors by AC susceptibility
measurements
Fedor Gömöry†
Institute of Electrical Engineering, Slovak Academy of Sciences, Dúbravská 9,
84239 Bratislava, Slovakia
Received 16 December 1996, in final form 2 May 1997
Abstract. A magnetic field harmonically varying in time (to probe the sample) and
a lock-in technique (to register the sample response sensed by a pick-up coil) are
widely used for characterizing superconductors. Measuring the temperature
dependence of the complex AC susceptibility is the most common procedure of this
type. This paper reviews these techniques, introducing in addition the complex AC
susceptibility, the so-called ‘wide-band AC susceptibility’. The latter quantity refers
to the magnetic flux and often offers an easier meeting between theory and
experiment. Starting from models for linear flux diffusion, reversible screening,
volume and surface flux pinning and the intermediate regime in a type II
superconductor, the expressions for the complex AC susceptibility in different
cases are presented and compared with those derived for the wide-band AC
susceptibility. Derivation of the basic physical properties of high-Tc
superconducting materials from the AC data (resistivity, critical temperatures and
fields, London and Campbell penetration depths, critical current density, granularity
and content of superconducting phase, irreversibility line, pinning potential) is then
thoroughly discussed.
1. Introduction
A magnetic field harmonically varying in time (to probe
the sample) and the lock-in technique (to register the
sample’s induced magnetic response sensed by a pick-up
coil) are often used to study the electromagnetic properties
of superconductors. Measurement of the temperature
dependence of the complex AC susceptibility is the most
common experiment of this type.
As we will see, a common theoretical background
exists for the experimental techniques that can be treated as
members of the same family: complex AC susceptibility,
AC losses, inductive measurement of Tc and critical
magnetic fields, AC methods to study pinning. This paper
is an attempt to systematize the knowledge gathered in
the field, with particular attention paid to the modifications
introduced after the discovery of high-Tc superconductors
(HTSCs).
A simple registration of the AC susceptibility is enough
to characterize the type I superconductor (Shoenberg
1937, Hein and Falge 1961). The development of the
type II superconductors required experimental methods to
study the volume pinning force and the surface barriers
† Present address: Pirelli Cavi SpA, c. 2714, Viale Sarca 222,
20126 Milano, Italy.
c 1997 IOP Publishing Ltd
0953-2048/97/080523+20$19.50 (Campbell 1969, Rollins et al 1974, Griffiths et al 1976).
AC susceptibility was used to reveal the filamentary
superconductivity (Maxwell and Strongin 1963) and effects
of granularity (Ishida and Mazaki 1979) and the importance
of correlating the AC susceptibility data with the structure
was pointed out (van der Klein 1972). The AC losses are
probably one of the most important topics related to the AC
susceptibility (Clem 1979a, Campbell 1995, Müller et al
1996). Recently the wide-band AC susceptibility technique
was developed (Dubots and Cave 1988, Gömöry 1991b,
Gugan 1994, Stoppard and Gugan 1995) that could serve as
an integrating concept for all these methods. It will be used
throughout the paper, exploiting such favourable features
as simple derivation of formulas and easy interpretation of
results.
The phenomenology of the macroscopic magnetic
properties of type II superconductors is nicely mapped in
the studies centred around the critical state model and the
concept of the critical current density (Campbell and Evetts
1972, Ullmaier 1975). After the discovery of HTSCs, all
the previous ‘low-Tc ’ experience was confronted with new
problems (Malozemoff 1989), in particular an extremely
rich phase diagram of the flux line lattice (Blatter et al
1994). Higher operational temperatures and strong intrinsic
anisotropy emphasize the importance of dynamic effects
(Brandt 1992a).
523
F Gömöry
Physical models used to explain the AC susceptibility
are identical with those utilized in DC magnetic
experiments (Senoussi 1992, Pérez et al 1996) and
comparison of both methods gives valuable information
(Frischhertz et al 1995). Other experimental techniques
that allow us to study similar properties are the vibrating
reed experiments (Köber et al 1991, Parvin et al 1993,
D’Anna et al 1994, Ziese et al 1994, Rogacki et al 1996)
and the AC transport measurements (Doyle R A et al 1994).
The basic arrangement of the experiment (a detailed
description can be found e.g. in the papers of Ramakrishnan
et al (1985), Rillo et al (1991), Couach and Khoder (1991)
and Nikolo and Hermann (1991)) is given in figure 1.
The sample in the form of a slab (large and high enough
for these dimensions to be considered infinite compared
with the sample width) is placed in the magnetic field
Bext = Bd +Bac having the static component Bd and the AC
component Bac characterized by the frequency f = ω/2π
and the amplitude Ba given by
Bac = Ba cos(ωt)
(1.1)
parallel to the surface of the slab. We suppose that the DC
and AC fields are parallel in space, although sometimes
the perpendicular arrangement of the fields is used (de la
Cruz et al 1994, Filippi et al 1994, Waldmann et al 1996).
The sample’s magnetic response is sensed by a pick-up coil
surrounding the sample. The temperature control is another
subsystem of the apparatus, allowing the dependence of the
sample’s parameters on temperature to be studied.
Various methods exist to analyse the pick-up coil
voltage: the whole waveform can be analysed (e.g. Cave
et al 1991, Gjomesli and Fossheim 1994b, Kerchner et al
1995), the waveform analyser can give the Fourier spectrum
of the signal (e.g. Fabbricatore et al 1993a, 1994, Mazaki
et al 1995) or, as the most simple way, only the effective
value is detected by an AC voltmeter. In this paper we
consider the phase-sensitive detection (PSD) that allows
us to measure AC susceptibility and seems to be a good
compromise between complexity of the data and simplicity
of the experiment.
The organization of the paper is as follows: in
section 2 the basic terms are introduced. Together with
the complex AC susceptibility components, the wide-band
AC susceptibilities will be defined.
To interpret the experimental data correctly we need
to recognize what mechanisms participate in the observed
behaviour. I gathered the formulas for both types of
AC susceptibilities derived according to the models of
linear flux diffusion (including flux flow and thermally
assisted flux flow), reversible screening (covering both
the Meissner–Ochsenfeld effect and Campbell’s reversible
motion of flux lines), critical state, surface barrier pinning
and flux creep in section 3.
Then, in section 4 the procedures that allow the critical
temperature, critical magnetic fields, linear resistivity,
critical current density, surface barrier and granularity
together with the content of the superconducting fraction,
London’s and Campbell’s penetration depths, irreversibility
line and so-called pinning potential (the height of barrier for
thermal activation), to be determined are recommended.
524
2. Definition of basic terms
The magnetic flux crossing the sample area can be
expressed with the help of the mean value B of the flux
density through the sample cross-section A:
Z
φm = AB =
B · dA.
(2.1)
A
In an AC experiment B is a function of time and controls
the voltage in one turn of the pick-up coil:
um (t) = −
dB(t)
dφm (t)
= −A
.
dt
dt
(2.2)
It is important to bear in mind that the pick-up coil measures
an integrated value of the flux density in the sample.
Recently experiments have been performed using miniature
Hall probes to record the local magnetic response of the
sample (Prozorov et al 1995, van der Beek et al 1996,
Morozov et al 1996), but these are outside the scope of
the present review. Using equation (2.1) one can define the
sample magnetization
M(t) = B(t) − Bext (t) =
φm (t)
− Bext (t).
A
(2.3)
The complex AC susceptibility components are defined as
(Maxwell and Strongin 1963)
1
χ =
π Ba
0
χ 00 =
1
π Ba
Z
2π
M(ωt) cos(ωt) d(ωt)
0
Z
2π
M(ωt) sin(ωt) d(ωt).
0
The physical meaning of χ 0 and χ 00 is the following.
The energy converted into heat during one cycle of the AC
field is (Clem 1988)
Wq = −2π χ 00
Ba2
.
2µ0
(2.4)
This expression explains why the lock-in can be used to
determine AC losses (Sekula 1971, Bozec et al 1991,
LeBlanc and LeBlanc 1992, Jiang and Bean 1994).
Because Wq is always negative, χ 00 in a correctly designed
experiment must take the positive sign.
The time average of the magnetic energy stored in the
volume occupied by the sample is (Gömöry 1991a)
Wm = χ 0
Ba2
2µ0
(2.5)
where the normal-state value was taken as the reference
level, i.e. Wm (T > Tc ) = 0. Diamagnetic behaviour leads
to reduction of the magnetic field compared with the normal
state, reflected in a negative value of Wm . Thus we expect
in the case of a superconductor χ 0 ≤ 0.
The wide-band susceptibilities are defined as (Gömöry
1991b).
Ma
Mr
χr =
(2.6)
χa =
Ba
Ba
Characterization by AC susceptibility
Figure 1. The set-up for AC susceptibility measurement on superconductors (schematic).
where Ma is the sample magnetization at the moment when
the external field reaches the maximum: we can call it
the ‘amplitude magnetization’. Mr is the magnetization
remaining in the sample at zero instantaneous value of the
AC field: we can call it the ‘remanent magnetization’ (see
figure 2). Therefore in the following χa and χr will be
called the remanent and the amplitude AC susceptibility,
respectively.
The diamagnetic nature of the superconductor leads to
a negative magnetization of the sample at the moment when
the external field reaches extremal (both negative and positive) values −Ba and Ba , respectively. The absolute value
of χa ∈ (−1, 0) is a measure of the external flux density
that is not allowed to penetrate the sample. If the flux density inside the sample has some delay with respect to the
external field, a non-zero remanent magnetization (i.e. left
in the sample at zero Bac ) appears. It is seen as a hysteresis
of the AC magnetization loop M(Bac ) and leads to χr > 0.
Regarding the instrumentation, complex AC susceptibility requires a lock-in with a harmonic reference waveform or, in the case of an instrument with a rectangular
reference waveform, to filter the input voltage. To record
the amplitude and the remanent AC susceptibility, we must
pass the entire spectrum of the pick-up coil signal to the
PSD with the rectangular reference waveform. If the pickup coil signal does not contain higher harmonics, exact
equivalence is reached: χ 0 = χa , χ 00 = χr .
Both χ 0 and χa reflect the shielding ability, while χ 00 as
well as χr are measures of the magnetic irreversibility. The
main difference from the practical point of view is in the
complexity of calculations: the complex AC susceptibility
is related to the whole magnetization cycle, while χa and
χr are determined at only two significant instants of the
cycle. This sometimes results in a simpler calculation of
χa and χr in comparison with χ 0 and χ 00 and consequently
easier comparison of the experimental results with theory.
3. Superconductor in a harmonic external field
The interaction between superconductor and magnetic field
consists of various processes. Those treated here are the
expulsion of flux due to Meissner–Ochsenfeld effect, the
flux pinning by a surface barrier, the pinning of the flux
entering the bulk of superconductor in the form of flux
lines, the diffusion of these flux lines across the sample
and the reversible motion of pinned flux lines in potential
wells. Analysing the physical models of these mechanisms
one can see that they can be grouped in the following way
(Brandt 1990). The diffusion equation (we denote by x the
transverse coordinate)
1
∂B
µ0 ∂B
∂ 2B
=
=
2
∂x
ρ(B, E) ∂t
D(B, E) ∂t
(3.1)
525
F Gömöry
Figure 2. One cycle of the magnetization of a YBaCuO melt-grown sample. Left: the time dependences of the driving field
and the sample magnetization. Right: the magnetization loop. The significant values of magnetization recorded when
Bac ± Ba and Bac = 0 will be used in the definition of the wide-band susceptibilities. Note the obvious anharmonicity of M (ωt ).
together with the approximate current–voltage characteristics (Vinokur et al 1991, Gilchrist 1994, Gilchrist and van
der Beek 1994)
j (B, E) = jc (B)
E
|E|
|E|
Ec
1/n
(3.2)
allow us to model the linear diffusion of flux lines (n = 1),
critical state (n → ∞) and intermediate regimes (n > 1).
The resistivity ρ(B, E) from equation (3.2) defines the
relation between the electrical field and the current density:
E(x, t)
.
j (B, E, x, t) =
ρ(B, E)
δ=
2ρlin
µ0 ω
1/2
.
(3.3)
where 3 is a material constant. This equation describes
well the reversible screening either by Meissner currents or
by Campbell’s reversible motion of flux lines in potential
wells.
Finally, the irreversible surface barrier must be treated
by a model derived from a simplified form of the
magnetization loop.
3.1. Linear diffusion
Setting n = 1 in equation (3.2) gives j linearly proportional
to E, i.e.
E
(3.1.1)
j (B, E) =
ρlin (B)
and only one material parameter ρlin (B) = Ec (B)/jc (B)
remains. Substituting equation (3.1.1) into equation (3.3)
converts equation (3.1) into a linear differential equation,
which leads to an AC flux profile with the local AC field
amplitude B0 (x) decaying roughly exponentially (Brandt
1991a):
(3.1.2)
B0 (x) ≈ Ba exp(−x/δ).
(3.1.3)
Important mechanisms of the flux dynamics belonging
to this case are normal-state eddy currents, linear flux flow
and thermally assisted flux flow. They differ in the value
of the linear resistivity. For eddy currents this is equal to
the normal resistivity:
ρlin = ρn .
For some mechanisms one has to use the London (1961)
equation
∇ × (3j) = −B
(3.4)
526
The characteristic space scale of the AC field decay is
(3.1.4)
When the Lorentz force fL = j B exceeds the pinning
strength of the material (characterized by the volume
pinning force fP ), the flux line lattice (FLL) starts to move
as a whole. Kim and Stephen (1969) found this movement
to be viscous, with the viscosity coefficient
ηF F =
BBc2
ρn
(3.1.5)
where Bc2 is the upper critical magnetic field. The flux
lines move with velocity vL proportional to the gradient
in flux density, and this movement is accompanied by the
electrical field E = B × vL (Josephson 1965). Thus, the
resistivity ρF F that appears in equation (3.1.3) is now called
the ‘flux flow resistivity’ (Kim and Stephen 1969):
ρF F = ρn B/Bc2 .
(3.1.6)
This becomes field independent when we use a large DC
field Bd superimposed on the AC field because then ρF F =
ρn Bd /Bc2 . Then, for linear flux flow,
ρlin = ρn Bd /Bc2 .
(3.1.7)
The FLL deformed by local pinning forces tends to
reach states with lower energy. This process could by
activated by several mechanisms, thermal activation being
Characterization by AC susceptibility
χ 00 and χr will be reached when δ = 0.887R in the case of
a cylinder (radius R) or δ = 0.556R in the case of a slab
(width 2R).
In section 4 we will see how difficult it is to cope
with a combination of several mechanisms appearing
simultaneously in the flux dynamics.
One possible
approach is to work out linearized models that can be
superimposed. Brandt (1992b) succeeded in deriving a
complex resistivity that combines the effects of Meissner
screening, thermally assisted flux flow, flux flow and
reversible motion of flux lines (Campbell 1971):
ρAC (ω) = iωµ0 λ2L + ρT AF F
Figure 3. AC susceptibility in the regime of linear flux
diffusion.
the most important. At low density gradients (Kes et al
1989), for thermally assisted flux flow,
ρlin = ρF F exp(−U/kB T ).
(3.1.8)
Here U is the typical height of the pinning barrier.
The linear diffusion is treated in textbooks of classical
electrodynamics (e.g. Smythe 1950, p 390). Because of
the linear character of the diffusion, higher harmonics are
not generated in the pick-up signal and χ 0 = χa as well as
χ 00 = χr . For a slab of width 2R in a parallel magnetic
field one finds (Kes et al 1989)
χ 0 = χa =
δ sinh(2R/δ) + sin(2R/δ)
−1
2R cosh(2R/δ) + cos(2R/δ)
δ sinh(2R/δ) + sin(2R/δ)
.
χ = χr =
2R cosh(2R/δ) + cos(2R/δ)
00
1 + iωτ
.
1 + iωτ0
(3.1.11)
Here λL is the London penetration depth and the time
constants are as follows: τ = αL /ηF F is given by
the Labusch parameter αL (Labusch 1969, Brandt 1991b,
Zeisberger et al 1994) and the flux flow viscosity ηF F ,
and τ0 = τ exp(−U/kB T ). Linear AC response theory
provides elegant and compact solutions (Campbell 1991)
and can account for the anisotropy (Wu and Tseng 1996).
The linear AC susceptibility could be used to reflect the
conductivity of the samples (e.g. Kötzler et al 1994, Ando
et al 1994, Li et al 1994), but one should always test
whether the conditions of linearity are fulfilled (Mehdaoui
et al 1993). However, many practical applications require
the determination of the superconducting properties in the
conditions of a strongly non-linear regime (Takács and
Gömöry 1995).
3.2. Critical state
(3.1.9a)
The behaviour of a superconducting material able to pin the
flux line lattice is well described by the so-called criticalstate model (Bean 1964). The j (B, E) dependence is given
by the expression (3.2) with n → ∞. The flux density
gradient in the critical state will be
(3.1.9b)
∂B
= ∓µ0 jc (B).
∂x
The expressions for a spherical sample (Khoder and Couach
1991) are of similar form. Another important geometry is
the case of a cylinder with radius R in a parallel magnetic
field. The corresponding formulas are (Clem et al 1976):
2J1 (kR)
0
−1
(3.1.10a)
χ = χa = Re
kRJ0 (kR)
2J1 (kR)
χ 00 = χa = Im
.
(3.1.10b)
kRJ0 (kR)
The penetration depth δ enters the expressions (3.1.9) and
(3.1.10) through an auxiliary variable k = (1 + i)/δ, i is
the imaginary constant and J0 , J1 are the Bessel functions
of the first kind (e.g. Spanier and Oldham 1987, p 509).
Re means the real part while Im is the imaginary part.
The graphical form of (3.1.9) and (3.1.10) is presented in
figure 3.
The linear regime is dissipative, which means that a
non-zero loss of magnetic energy appears and the peak in
(3.2.1)
Accordingly, the material is characterized by the critical
current density jc .
Suppose the dependence of jc on the AC magnetic field
can be omitted. This approximation is valid if for example
Bd Ba because jc (B) ≈ jc (Bd + Ba ). At the moment
when the AC magnetic field (1.1) reaches a maximum value
(i.e. at ωt = 0, 2π , 4π, . . . , 2nπ ), the AC flux density
profile takes the form
B(x, 0) = Ba − µ0 jc x.
(3.2.2)
Field penetration is stopped in the depth
xc =
Ba
.
µ0 jc
(3.2.3)
Because of the strong non-linearity of the current–
voltage relation, higher harmonics will appear in the sample
magnetization causing anharmonicity in the pick-up coil
voltage (Fabbricatore et al 1994). As a consequence, χr is
no longer equal to χ 00 or χa to χ 0 . In the following formulae
527
F Gömöry
we use an auxiliary variable y = xc /R = Ba /µ0 jc R. For a
slab (width 2R) in a parallel magnetic field (Ji et al 1989,
Matsushita and Ni 1989, Goldfarb et al 1991)

y/2 − 1
for 0 ≤ y ≤ 1


 

 1
y
− 1 cos−1 (1 − 2y)
χ0 =
π
2



4
4


 + −1+ − 2 (y −1)1/2 for 1 ≤ y
3y 3y
(3.2.4a)

2y/3π
for
0
≤
y
≤
1

(3.2.4b)
χ 00 =
4
1 6

− 2
for 1 ≤ y
3π y
y
and the wide-band susceptibilities take a form that can be
easily derived from the flux density profiles (Bean 1964):
(
y/2 − 1
for 0 ≤ y ≤ 1
(3.2.5a)
χa =
−1/2y
for 1 ≤ y


 y/4
χr = 1 − y/4 − 1/2y


1/2y
for 0 ≤ y ≤ 1
for 1 ≤ y ≤ 2
(3.2.5b)
for 2 ≤ y.
If the sample has the form of a cylinder with the radius R,
then in a similar way as for the slab we find
 5y


y
1
−
−1
for 0 ≤ y ≤ 1


16


 2
5y 2
χ0 =
−1+y −
cos−1 (1−2y)

π 16




2
19 5y 1
1/2

 +− + + −
(y −1)
for 1 ≤ y
12 8 y 3y 2
(3.2.6a)

4


y(1 − y/2)
for 0 ≤ y ≤ 1

3π (3.2.6b)
χ 00 =
1
4 1



1−
for 1 ≤ y
3π y
2y
(
y − y 2 /3 − 1
for 0 ≤ y ≤ 1
χa =
(3.2.7a)
−1/3y
for 1 ≤ y

2
for 0 ≤ y ≤ 1

 y/2 − y /4
2
χr = −1/3y + 1 − y/2 + y /12
for 1 ≤ y ≤ 2


1/3y
for 2 ≤ y.
(3.2.7b)
These dependences are presented in figure 4. It is worth
mentioning some important features.
(1) When xc R (i.e. y → 0), χa ∼
= χ and
∼
χr = (3π/8)χ 00 for both geometries.
(2) The penetration of the AC field is dissipative and
the persistent currents cause the appearance of the remanent
00
magnetization. The peak
√ in χ as well as in χr is reached
at xc = R and xc = 2R in the case of cylinder and slab,
respectively.
(3) There is a notable difference in the heights of the
maxima between χ 00 and χr for both geometries.
(4) The substantial difference between χa and χ 0 for
xc R indicates that the higher harmonics significantly
528
Figure 4. AC susceptibility in the regime of flux pinning in
the sample bulk, described by the critical-state model.
contribute to the pick-up coil signal. This can be used to
detect the onset of the critical state by the appearance of
the third harmonics in the susceptibility signal (Gilchrist
and Konczykowski 1990, Yang et al 1994b, van der Beek
et al 1995, 1996, Dubois et al 1996).
The portion of the χr (xc /R) and χa (xc /R) curves when
xc > 2R can be used very simply for the determination
of the critical current density. In the case of a slab-like
sample (width 2R) we find by combining equation (3.2.3)
with equations (3.2.5) that
jc =
2χa Ba
2χr Ba
=−
µ0 R
µ0 R
for jc <
Ba
2µ0 R
(3.2.8)
and, in a similar way, combining equation (3.2.3) with
equations (3.2.7) we find that for a cylindrical sample with
radius R
jc =
3χa Ba
3χr Ba
=−
µ0 R
µ0 R
for jc <
Ba
.
2µ0 R
(3.2.9)
These expressions can be used to transform directly the
measured χ(T ) curves into the jc (T ) plot. Only wideband susceptibilities provide this opportunity because of the
linear form of the relations (3.2.8) and (3.2.9) in the interval
xc > 2R. The use of χ 00 (T ) for this purpose (Senoussi
1992) requires the more restrictive condition xc R to be
fulfilled. The critical-state model was found to explain the
behaviour of many types of HTSC samples (Berling et al
1996b).
3.3. Intermediate regimes
This is the case of equation (3.2) when n is greater than 1
but not enough to be considered infinitely large. Therefore
it is called the intermediate regime (Civale et al 1991,
Karkut et al 1993, Polichetti et al 1994, DiGioacchino et al
1996). Typical examples of this behaviour are the extreme
Characterization by AC susceptibility
flux flow (Takács and Gömöry 1990) and the flux creep
(Anderson and Kim 1964) regimes.
The extreme flux flow model was worked out to treat
the flux flow at Bd = 0, when because of the flux flow
resistivity dependence on the magnetic field (3.1.6) the
diffusion equation (3.1) is no longer linear. An approximate
solution for the penetration depth was found, i.e.
3πBa ρn 1/2
δEF F =
(3.3.1)
8µ0 ωBc2
that nicely demonstrates the characteristic feature of the
intermediate regimes: the penetration depth depends on
both the AC field amplitude as well as the frequency.
Careful experiments we performed (Takács and Gömöry
1995) demonstrated that the regime of this type with n = 2
can be found in melt-grown YBaCuO samples near Tc .
In contrast, the flux creep covers a wide range of
temperatures and fields in the case of HTSCs. This regime
is met when in the sample, driven originally into the
critical state, separate pieces of the FLL (flux line segments,
bundles of flux lines, etc.) are hopping over pinning
barriers to reach metastable states with lower energy, and
the movement against the density gradient can be neglected.
Symptomatic of thermally activated creep is the appearance
of a drift velocity of the flux lines vL that obeys the
Arrhenius law:
Ueff
vL = v0 exp −
.
(3.3.2)
kB T
Here, Ueff is the effective height of the barrier that the
hopping segment of the FLL must overcome, kB = 1.38 ×
10−23 J K−1 is the Boltzmann constant, and v0 is the line
velocity at Ueff → 0 that can be estimated considering the
transition from flux creep to flux flow (van der Beek et al
1992).
The flux creep is often taken as a small perturbation in
the pinning mechanism (e.g. Fedorov and Stepanov 1996).
Then, in equation (3.2), jc is the unperturbed value of the
current density given by the flux pinning strength while the
exponent
∂ ln E
n=
(3.3.3)
∂ ln j
expresses the significance of relaxation mechanisms. On
decreasing n we can model situations with increasing
importance of the flux creep (Fabbricatore et al 1996).
The dynamic penetration depth for the combination of
pinning and creep x0 is found for example by replacing jc
in equation (3.2.3) by j given by equation (3.2):
x0 =
Ba
Ba
=
.
µ0 j
µ0 jc (E/Ec )1/n
(3.3.4)
The characteristic electrical field the sample experiences in
our experiment is (Campbell 1991)
E≈
R
ωBa .
2
(3.3.5)
Inserting this quantity into equation (3.3.4) one finds
1−1/n
x0 =
Ba
.
µ0 jc (Rω/2Ec )1/n
(3.3.6)
Figure 5. AC susceptibility in the regime of reversible
screening.
We expect that the maximum of χ 00 and χr will
be reached when x0 ≈ R (Geshkenlein et al 1991).
At high values of n, the penetration depth (3.3.6) is
only weakly frequency dependent while for n = 1 the
amplitude dependence vanishes. Measurements at different
frequencies (x0 ∼ ω−1/n ) could be used to find n.
3.4. Reversible screening
In this regime the sample passes through the equilibrium
states. This leads to a reversible character of the sample
magnetization, yielding χ 00 = χr = 0. The pick-up coil
signal does not contain higher harmonics and χ 0 = χa .
From equation (3.4) it follows that the magnetic field
decreases exponentially with the characteristic length λ =
(3/µ0 )1/2 .
The susceptibilities are (Campbell et al 1991)
R
λ
0
χ = χa = − 1 −
tanh
(3.4.1)
R
λ
for a slab of width 2R in a parallel field and
2λ I1 (R/λ)
0
χ = χa = − 1 −
R I0 (R/λ)
(3.4.2)
for a cylinder of radius R in a parallel field. In the
formula for the cylinder, the modified Bessel functions I0
and I1 , sometimes called the Neumann functions (Spanier
and Oldham 1987, p 533), are used.
The graphical form of the susceptibilities is given in
figure 5. Two important points are to be noted.
(1) The temperature dependence of the penetration
depth can be studied using AC susceptibility data measured
on an object with well-defined geometry. The highest
sensitivity of the data on the changes in λ is reached for
R ≈ λ.
529
F Gömöry
(2) A material of thickness comparable with λ is far
from completely expelling the magnetic flux in spite of the
fact that it is perfectly superconducting. This effect, called
‘magnetic invisibility’ (Clem and Kogan 1987), could lead
to severe underestimation of the superconducting phase
content particularly in the case of BiSrCaCuO compounds
(Plecháček and Gömöry 1990).
There are two typical examples of reversible mechanisms entering the flux dynamics: screening by Meissner
currents and Campbell’s reversible screening.
Meissner currents try to expel the magnetic flux from
the superconducting volume to reach thermodynamical
equilibrium. They are maximal on the interface with the
external magnetic field; the characteristic space scale is the
London penetration depth λL . The maximum drop of the
field across the sample they can produce is equal to Bc1 ,
the lower critical magnetic field. At higher fields these
currents only weaken the external field Bext to the actual
value entering the sample bulk B = Bext − Bc1 . Both
Bc1 as well as λL are temperature dependent, the values at
T → 0 being among the basic physical characteristics of
the material.
In the experiment we can neglect the Meissner currents
either when the magnetic fields are much larger than Bc1 or
when there is no structure in the sample with dimensions
comparable with λL (Ishida and Mazaki 1987). Taking into
account that the values of λL are in the range of 1000 Å one
can expect that the Meissner currents could be important for
micrometre-sized samples.
In the mixed state, we can imagine the individual flux
lines to be positioned in potential wells. At small driving
force, the displacement of the flux line from the bottom of
the potential well could be reversible. Because of mutual
repulsion between flux lines, compression of the flux line
lattice appears starting from the surface. The deformation
vanishes in the depth
1/2
c
(3.4.3)
λC =
αL
given by the elastic modulus of the lattice c and the Labusch
parameter αL . In the case of Bd kBac the compressional
modulus c11 should be applied, which for B Bc1 is
material independent; c11 ≈ B 2 /µ0 where B 2 ≈ Bd2 + Ba2 .
The Labusch parameter characterizes the steepness of the
potential wells for the flux lines (Seow et al 1995). It is
defined as the mean value of the second derivative of the
potential energy U l ascribed to the flux line of unit length
in the field of pinning forces:
Z 2 l
∂ U
1
dA.
(3.4.4)
α=
A A ∂x 2
Indications of this effect have been found (Matsushita
et al 1992) and force–displacement curves derived from
the AC magnetization data (Campbell 1971, Johnson et al
1994, Johnson and Campbell 1996).
3.5. Surface pinning
In the previous considerations we supposed that there is
no barrier to prevent the flux lines entering or leaving the
530
Figure 6. AC susceptibility in the regime controlled by the
surface pinning. The magnetization loop of a sample with
the irreversible surface barrier Bb for the flux entry and exit
is plotted in the insert.
sample. However, the existence of such a barrier was
necessary to explain certain experiments on low-Tc samples
(Bean and Livingston 1964, Dunn and Hlawiczka 1968,
Melville 1972). Here we present a simple model (Cesnak
et al 1984) that is equivalent to the existence of a layer with
thickness δb and extremely high critical current density jb
on the surface of the sample. The relevant quantity is the
irreversible surface barrier
Bb = µ0 jb δb .
(3.5.1)
The magnetization loop of a sample with the irreversible
surface barrier is given in figure 6 (Clem 1991). The
susceptibilities do not depend on the sample geometry, and
one obtains (Clem 1979b)
χ0 =
1
(2 − sin 2 cos 2)
π
(3.5.2a)
4u
π(1 − u)
(3.5.2b)
χ 00 =
where u = Bb /Ba and cos 2 = 1 − 2u.
The wide-band susceptibilities can be calculated
directly from the magnetization loop with the results
(
−1
for 0 ≤ Ba ≤ Bb
(3.5.3a)
χa =
1 − Bb /Ba
for Bb ≤ Ba


0
χr = 1 − Bb /Ba


Bb /Ba
for 0 ≤ Ba ≤ Bb
for Bb ≤ Ba ≤ 2Bb
for 2Bb ≤ Ba .
(3.5.3b)
Characterization by AC susceptibility
These formulae are presented in graphical form in figure 6.
Both χ 00 and χr reach a maximum at Ba = 2Bb .
Notable differences between the complex AC susceptibility
components χ 0 , χ 00 and the wide-band susceptibilities χa ,
χr indicate a rich content of higher harmonics in the sample
magnetization.
We see that Ba /Bb plays in this mechanism the role
that is equivalent to for example the xc /R ratio in the case
of bulk pinning. In order to achieve a formal analogy we
can define the penetration depth as
xb = R
Ba
.
Bb
(3.5.4)
3.6. Identifying the regime that controls the flux
dynamics
A common feature of the mechanisms discussed in this
section was the existence of a characteristic space length
xp . When xp < R
(1) the time variation of fields and currents takes place
in the surface layer with thickness xp and
(2) the contribution to the magnetization of the sample
due to the currents circulating deeper than xp is negligible.
As far as χp depends on at least one of the external
parameters (steady magnetic field Bd , the AC field
amplitude Ba , the AC field frequency f = ω/2π and the
temperature T ), we have the possibility to change xp or,
more precisely, the ratio xp /R during the experiment. In the
neighbourhood of the value xp ≈ R, we expect to observe a
steep change of χ 0 or χa that will in the case of a dissipative
mechanism be accompanied by a peak in χ 00 or χr . In a
multiphase sample, several steps in χ 0 or χa corresponding
to several peaks in χ 00 or χr should be found (e.g. Couach
et al 1988, Cesnak 1992, Xia et al 1993). Changes of
the sample quality due to aging, contamination etc lead to
changes of the observed transition (Mehdaoui et al 1989)
To distinguish between different mechanisms, we must
check how this pattern is influenced by the controllable
parameters of the experiment (Yamaguchi et al 1994).
A summary of the essential features exhibited by the
mechanisms considered in this paper is given in tables 1
and 2.
4. Determination of a superconductor’s
parameters
The scope of this paper is to review the procedures allowing
us to extract as much information about the sample material
as possible from the AC data (summarized in table 2). With
the background information given in the preceding sections
we can deal with this task.
4.1. Linear resistivity (ρlin ) and critical temperature
(Tc )
In section 3.2 we have seen that in the case of linear
diffusion the resistivity ρlin is the only material parameter
controlling the flux dynamics. Then it can be found from
Figure 7. Dependence of AC susceptibilities (both
wide-band and complex AC susceptibility components) on
the superimposed DC field can be used to determine the
critical magnetic fields. Perfect reversible screening is
expected for Bdc < Bc 1 while the superconducting response
completely vanishes at Bdc > Bc 2 . Uncertainty in
determining the critical fields is roughly given by the AC
field amplitude.
equation (3.1.9) or (3.1.10), e.g. numerically (Polichetti
et al 1996).
To test whether the data used to determine ρlin
were taken in the linear regime we can compare the
susceptibilities measured at different amplitudes (to be sure
that they are amplitude independent) or various frequencies
(the resulting ρlin should be frequency independent)
or check the coincidence between the complex AC
susceptibility and the wide-band susceptibilities (to confirm
the non-existence of higher harmonics). In the case of
known Bc2 and normal-state resistivity, one can derive the
value of the typical height of the pinning barrier, U , from
the susceptibility measured in the TAFF regime.
The transition from a normal to a superconducting state
is always accompanied by a resistivity drop. Therefore an
AC susceptibility measurement can be used to determine the
critical temperature, Tc (e.g. Khan et al 1994, Masini et al
1994, Das and Suryanarayanan 1995). Sometimes the onset
temperature of transition, Ton , is used to characterize the
transition temperature and thus the quality of the sample.
In granular samples, however, Ton is often connected with
the Meissner shielding, resulting in a decrease of χ and χa ,
while χ 00 and χr remain zero (Opagiste et al 1994).
4.2. Critical magnetic fields Bc1 , Bc2
The upper critical magnetic field Bc2 plays, on the flux
density scale, a role similar to that of Tc on the temperature
scale: crossing of Bc2 should be accompanied by a
sharp change in the linear resistivity. Therefore the AC
susceptibility data can be used to find Bc2 by analysing how
the superimposed DC field shifts the onset temperature in
the linear regime (Küpfer et al 1987). This is schematically
illustrated in figure 7. Nevertheless, probably because of
the existence of other more exact methods, the AC data are
rarely used to determine Bc2 (Vlakhov et al 1994).
In the case when the lower critical field Bc1 is to
be determined, the situation is different. At low fields
and/or low temperatures the Meissner currents secure the
complete screening of a bulk sample without any magnetic
531
F Gömöry
Table 1. Qualitative characteristics of the mechanisms controlling the flux dynamics.
xp dependence on parameter
Mechanism
Formula for xp
T
Ba
ω
Bd
Higher
harmonics
Eddy currents
Linear flux flow
Thermally assisted
flux flow
Bulk pinning
Flux creep
Meissner current
Campbell’s reversible screening
Surface pinning
(3.1.3) + (3.1.4)
(3.1.3) + (3.1.7)
Weak
Strong
No
No
Strong
Strong
Weak
Strong
No
No
(3.1.3) + (3.1.8)
(3.2.3)
(3.3.6)
λL (London)
(3.4.3)
(3.5.4)
Strong
Strong
Strong
Weaka
Strong
Strong
No
Strong
Weak
No
No
Strong
Strong
No
Weak
No
No
No
Strong
Weak
Weak
No
Strong
Weak
No
Yes
Yes
No
No
Yes
a
Strong near Tc .
Table 2. Significant points on the AC susceptibility curves.
xp /R
at peak of χ 00 and χr
Model
Figure
Linear diffusion
Critical state
Flux creep
Reversible screening
Surface barrier
3
4
—
5
6
Cylinder,
radius R
0.556
1
∼1
—
2
irreversibility: χ 0 = χa = −1, χ 00 = χr = 0. Crossing the
Bc1 (T ) line towards higher temperatures or fields destroys
this perfect screening (Goldfarb and Clark 1985). It is easy
to find that
χa = −1
χa = −
Bc1
Ba
for Ba < Bc1
for Ba ≥ Bc1
(4.2.1)
χr = 0.
A more sophisticated method based on a dynamic analysis
of the flux line lattice in the reversible regime was proposed
by Khoder et al (1991). When the flux lines enter
the sample bulk they interact strongly with the material.
One can expect the appearance of dissipation as well as
incomplete diamagnetism when crossing the Bc1 (T ) line
and an upper limit for Bc1 (T ) can be determined in this way
(Goldfarb et al 1987b, Babic et al 1987, Loegel et al 1990,
Loughran and Goldfarb 1991). Gömöry and Takács (1997)
found that in the presence of flux pinning (characterized
by the critical current density jc ) it is still possible to find
the conditions allowing the determination of Bc1 , because
at large enough AC field amplitudes
χ a + χr = −
Bc1
Ba
for Ba ≥ 2µ0 jc R + Bc1 . (4.2.2)
In this way the temperature dependences χa (T ), χr (T ) can
be used to find Bc1 .
532
Slab,
width 2R
0.887
1.414
∼ 1.414
—
2
xp /R
at χ 0 ≈ χa ≈ −0.5
Cylinder,
radius R
0.556
0.634
∼ 0.634
0.301
2
Slab,
width 2R
0.887
1
∼1
0.522
2
4.3. Critical current density jc
Let us first analyse the simplest case when the flux pinning
is the only mechanism controlling the flux dynamics,
and the current-carrying capability is supposed to remain
constant throughout the sample. A rough but simple
estimation of jc can be performed by realizing that the
maximum of χ 00 or χr is reached at a particular value of
the AC field penetration depth: in the case of a cylinder
(Clem 1988)
Ba = µ0 jc R
at max{χ 00 } or max{χr }
(4.3.1a)
and in the case of a slab with thickness 2R (Gömöry 1989)
√
at max{χ 00 } or max{χr }. (4.3.1b)
Ba = 2µ0 jc R
Measuring at different AC field amplitudes Ba leads to
finding the peaks at different temperatures T and with the
help of equations (4.3.1) the jc (T ) dependence can be found
(Gömöry and Lobotka 1988, Mezzetti et al 1994, Widder
et al 1995).
The drawback of this approach is that it uses only
one point from the whole susceptibility curve. It is more
effective to use the expressions (3.2.5) or (3.2.7) and to
process a larger portion of the susceptibility data. Both
χa and χr must give the same jc , and the parts of the
jc (T ) curve determined from the data taken at different AC
field amplitudes must overlap (Lera et al 1992, Fábrega
et al 1993). Figures 8 and 9, respectively, illustrate
how wide-band AC susceptibility data can be used to plot
directly jc (T ) and jc (B) dependences. A slight difference
Characterization by AC susceptibility
Figure 9. (a ) Wide-band AC susceptibility of a YBaCuO
melt-grown sample as a function of the DC background
field, recorded at T = 84 K. The selected points were used
to construct the jc (B ) curve in (b ).
Figure 8. (a ) Wide-band AC susceptibility of a YBaCuO
melt-grown sample. The marked portion of the curves was
utilized to construct the jc (T ) curve in (b ). To allow the
comparison, the AC field amplitudes and frequencies were
chosen to induce the same electrical field on the sample’s
surface.
is observed between the jc values derived from χa and
χr . I suppose that this is due to periodic changes of
the electrical field that reaches a maximum during the
cycle just at the moment when the remanent magnetization
is determined, while the determination of the amplitude
susceptibility happens when the actual E ∼ dB/dt is zero.
Therefore the jc derived from χr are higher and represent
the sample’s current-carrying capability better.
Until now we have supposed that the magnitude of
jc does not change during the cycle. A possible check
of validity of this condition is the comparison of the
experimentally found peak height of χ 00 or χr with the
theoretical values given in figure 4. However, deviations
from these values can have other origins, the most important
being: reversible motion of flux lines (Matsushita et al
1991) and material granularity (Clem 1988). The value of
the critical current density can vary for various reasons.
Here we discuss two cases: jc changing as a result of
variations in the local value of magnetic field and jc
changing as a result of sample spatial inhomogeneity.
Suppose first that the properties of the sample are the
same everywhere and no DC field is applied. Then the
value of the magnetic field that affects the sample ranges
from 0 to Ba . Chen et al (1989) derived the formulae
for both components of the complex AC susceptibility by
considering Kim’s type of jc (B) dependence (Kim et al
1964): jc ≈ (B + B0 )−1 where B0 is a constant. They
succeeded in finding in this way the jc (T ) dependence for a
polycrystalline YBaCuO sample from the AC susceptibility
00
) was
data. A simpler analytical solution found for T (χpeak
used by Lee and Kao (1995) to interpret a large set of
data. Numerical calculations were performed to calculate
the complex AC susceptibility components for different
types of jc (B) dependences (Chen and Sanchez 1991a,
Forsthuber and Hilscher 1992, Sanchez 1994). For the
dependence
B −β
(4.3.2)
jc (B) = jc0 1 +
B0
having three parameters jc0 , B0 and β (where in particular
β = 0 gives constant jc and β = 1 corresponds to Kim’s
533
F Gömöry
model), the complex AC susceptibilities χ 0 and χ 00 were
found analytically in the approximation Bd Ba (Sun
et al 1995b) as well as the wide-band susceptibilities χa
and χr (Gömöry et al 1993a). For β > 0 the height of
the χ 00 or χr maximum exceeds the values illustrated in
figure 4, calculated for constant jc .
The necessity to consider jc (B) can be always avoided
by application of a large DC field. In contrast, Bd
comparable with Ba causes the effects created rather by a
non-linear interaction between both of the fields, masking
the intrinsic properties of the material (Gömöry et al 1989,
Gjomesli and Fossheim 1994a, Vuong 1996).
Another case is a macroscopically inhomogeneous
sample.
Here I do not mean the granularity of
polycrystalline samples which will be analysed in
section 4.5. The consequences of a broad distribution of the
critical temperatures in an Ag–BiSrCaCuO composite were
explained by Larrea et al (1994); sometimes the FWHM of
χ 00 peak is reported as 1Tc (Nafidi and Suryanarayanan
1995). A powerful AC magnetic method to study the
spatial inhomogeneity modelled by a sample consisting
of cylindrical shells with different jc was developed by
Campbell (1971). It is based on the evaluation of the mean
value of the flux density inside the sample at the moment
when the external field reaches maximum, B amp , and its
change with AC field amplitude. As the amplitude Ba
increases, the AC field probes successively more and more
internal shells of the sample. The depth δAC , to which the
AC field has penetrated, was found to be in the case of a
slab (thickness 2R)
δAC = R
dB amp
dBa
(4.3.3)
and in the case of a cylinder (radius R)
dB amp 1/2
.
δAC = R 1 − 1 −
dBa
(4.3.4)
Taking into account the definition of the amplitude AC
susceptibility (2.6), the χa data can be used for the
determination of δAC by replacing dB amp /dBa by d[(1 +
χa )Ba ]/dBa because B amp = Ba + Ma = Ba (1 + χa ). An
example of a graph obtained when applying Campbell’s
method is given in figure 10. The penetration depth δAC
increases with Ba , and the local critical current density
controls this proportionality:
jc (δAC ) =
1 dBa
.
µ0 dδAC
(4.3.5)
This means that the local critical current density in the shell
at a distance δ0 from the sample surface is proportional to
the slope of the Ba (δAC ) curve in figure 10 at δAC = δ0 .
In the case of a homogeneous sample with constant jc the
Ba (δAC ) curve should be a straight line.
4.4. Surface barrier Bb
In the case when the surface pinning is the only mechanism
controlling the flux dynamics, values of the surface barrier
Bb can be found by fitting the experimental data to
534
Figure 10. Campbell’s plot (‘flux density profile’) for a
YBaCuO single crystal with higher jc on the
well-oxygenated surface than in the interior.
the theoretical expressions (3.5.2). However, one must
normally consider at least the existence of volume pinning,
characterized by the critical current density jc . Clem
(1979b) derived the flux density profiles that can be used
for calculation of the wide-band susceptibilities in this case
(Gömöry and Takács 1997).
Complicated expressions are expected if one includes
the dependence of jc and Bb on the magnetic field.
However, this is very desirable because the role of
surface pinning seems to be more important than generally
supposed yet (Chen and Sanchez 1992, Sun et al 1994b,
Mamsurova et al 1994, Ding et al 1995).
4.5. Granularity and content of superconducting phase
(vs )
In the initial period of HTSC development, all the prepared
samples were polycrystalline, exhibiting behaviour explainable by the model of a ‘better’ phase embedded in a ‘worse’
matrix (Goldfarb et al 1987a). A typical manifestation of
such behaviour is the AC susceptibility transition with two
steps in χ 0 (T ) accompanied by two peaks in χ 00 (T ) as illustrated in figure 11 as well as the Campbell graph composed
of two lines with different slopes (figure 12). It was early
recognized that such behaviour is expected in the case of
a sample consisting of superconducting grains representing
elementary volumes of the phase coherence of the order parameter, interconnected by a system of weak links (Ishida
and Mazaki 1987, Küpfer et al 1988, Müller 1990, Senoussi
et al 1991, Levy et al 1994). When cooled under Tc the
grains become superconducting first, and shielding is performed by the so-called intragrain currents with density jg .
On lowering the temperature further, locking of the order
parameter phases in different grains is reached (Clem 1988)
and the whole sample behaves like a material able to carry
the macroscopic intergrain current density jm , defined as
P
Atot II G,i
(4.5.1)
jM =
Atot
where the intergrain currents II G are summed flowing
through a plane parallel to Bext and containing the principal
symmetry axis: see figure 13. Such a definition does not
give the actual current density carried by the intergrain
contacts locally, but is equivalent to that determined in
Characterization by AC susceptibility
Figure 11. AC susceptibility curve of a polycrystalline
YBaCuO sample with pronounced granularity.
Figure 13. (a ) Schematic drawing of the paths of the
intragrain current density jg and the (macroscopic)
intergrain current density jm . (b ) The intergrain current
density is related to the macroscopic area Atot .
Figure 12. Campbell’s plot (‘flux density profile’) for the
polycrystalline YBaCuO sample with pronounced
granularity.
the contact transport measurements. Then one expects to
observe a dependence of the susceptibility on the magnetic
history of the sample (Gömöry et al 1989, Puig et al
1992, Dhingra et al 1995). Probably the most complete
explanation of this effect was given by Saha and Das
(1993).
By enhancing the quality of the intergrain contacts, the
difference between inter- and intragrain currents vanishes,
and the two steps (and peaks) merge (Sauv et al 1993,
Planinić et al 1994, Vo et al 1996). However, very small
fractions (∼1%) of a poorer quality superconductor or the
weak links in single crystals could be detected (Moreira
et al 1994, Ren et al 1994) by AC susceptibility.
Let us demonstrate the way of extracting essential
information from the AC susceptibility data taken on a
granular sample. We start with the simplest model: inside
the grains the flux dynamics is governed by the intragrain
pinning force characterized by the intragrain critical current
density jgc while the system of intergrain contacts forms
an effective pinning medium (Clem 1989, Müller et al
1991, Ravi and Seshu Bai 1994, Doyle T B et al 1994,
Feigelman and Ioffe 1995, Chen et al 1995), characterized
by the intergrain critical current density jmc . Because
of the smaller dimensions of grains with respect to the
whole sample and because generally jgc is larger than jmc ,
the χ 00 (T ) peak appearing at higher temperature (T1 in
figure 11) has to be ascribed to the situation when the AC
field penetrates just to the centres of grains. The critical
state model (section 3.2) allows us to determine
jgc |T =T1 =
Ba
µ0 Rg
(4.5.2)
where Rg is the grain radius. For simplicity we suppose that
all the grains are identical and the sample is dense enough to
exhibit strong magnetic coupling between individual grains
(Senoussi et al 1990). Then the system of grains can
be handled as a set of cylinders parallel to the external
field (Müller 1989). Questionable is here the estimation of
the grain size. Early studies revealed that the elementary
superconducting coherent volume can differ from the grain
size observed for example by optical methods (Däumling
et al 1988, Chen et al 1990, Nicolas et al 1995) and the
535
F Gömöry
existence of intragrain contacts was suggested (Küpfer et al
1989).
An analogous problem does not exist in the determination of jmc from the χ 00 (T ) peak observed at lower
temperature (T2 in figure 11). This peak is reached when
the intergrain currents arrive at the value allowing the AC
field to penetrate through the intergrain space just to the
sample centre. Then, for a cylinder (radius R) and a slab
(thickness 2R) respectively
jmc |T =T2 =
Ba
µ0 R
Ba
jmc |T =T2 = √
.
2µ0 R
(4.5.3)
Using a series of AC field amplitudes, a set of peak
temperatures T2 is obtained and the jmc (T ) dependence can
be constructed.
When the temperatures T1 , T2 are sufficiently distant
the peaks can be handled separately. Then the models
valid for a homogeneous superconductor, i.e. involving
more mechanisms, can be applied to the portions of data
manifesting the inter- and the intragrain currents (Chen and
Sanchez 1991b, Sanchez and Chen 1991).
When trying to reveal the existence of links controlling
the intergrain currents, Campbell’s method is very useful.
At low amplitudes Ba the field is supposed to penetrate first
the intergrain volume, and therefore the initial part of the
Ba (δAC /R) curve corresponds to the shielding performed
by the intergrain currents. At a certain value Ba∗ the
AC field is just sufficient to occupy the whole intergrain
space and the AC flux starts to enter the grains. The
flux penetration into grains at B > Ba∗ is opposed by the
intragrain current density and the part of the Campbell curve
at B > Ba∗ can be used to determine jgc . Thus, analysing
the graph from figure 12 one finds
Ba∗
1
dBa jgc =
. (4.5.4)
jmc =
µ0 R
µ0 Rg d(δAC /R) B>Ba∗
At B = Ba∗ the whole intergrain space is occupied by
the AC field. Because the total volume of the sample Vtot is
the sum of the intergrain space Vi and the space occupied
by the superconducting grains Vg , one finds from simple
geometrical considerations that the ‘porosity’ of the sample
can be defined for a slab (thickness 2R) and a cylinder
(radius 2R) as respectively

δAC ∗



 R
Vi
= (4.5.5)
vi =
2

Vtot
δAC ∗



.
R
The same result is obtained when we suppose that the grains
are perfectly shielded by the intragrain Meissner currents
at Ba ≈ Ba∗ .
Let us derive the wide-band AC susceptibilities in the
following model case (Opagiste et al 1993): the grains
are perfectly diamagnetic in the considered interval of
temperatures and AC field amplitudes owing to Meissner
currents, while the intergrain currents shield the sample as
a whole, creating the critical state described by the critical
current density jmc (Bekeris et al 1994). For simplicity
536
we neglect the possible coupling between the inter- and
intragrain currents (Kasatkin et al 1994). The diamagnetic
action of the intragrain currents excludes the sample volume
fraction fs from the penetration as well as capturing the AC
magnetic flux (Calzona et al 1989, Marohnić and Babic
1991). This leads to the modification of formulae (3.2.5)
or (3.2.7). Then, for a cylinder (radius R) at Ba > 2µ0 jmc R
χa = −
µ0 jmc R
− fs
3Ba
χr =
µ0 jmc R
(1 − fs ) (4.5.6a)
3Ba
and in the case of a slab (thickness 2R)
χa = −
µ0 jmc R
−fs
2Ba
χr =
µ0 jmc R
(1−fs ). (4.5.6b)
2Ba
This means that the Meissner shielding of grains will
depress the height of the intragrain χr peak and in the same
time sharpen the χa step. Data of this type were found on
a Bi-2212 melt-grown sample (Gencer et al 1996a) and a
Bi-2223 polycrystal (Gencer et al 1996b).
The case of large AC amplitudes or low intergrain
currents is important, when
χa |Ba µ0 jcm R = −fs
χr |Ba µ0 jcm R = 0.
(4.5.7)
This means that if a ‘valley’ in χr exists between T1 and
T2 that goes down to zero, the corresponding value of
χa gives directly the diamagnetic fraction of the sample.
Too weak an AC field amplitude would disable this type
of determination because a minor superconducting fraction
could also shield the whole sample against a low AC field
(Dubois et al 1995).
Alternatively, fs can be derived from the height of the
intergrain maximum (i.e. that found at T2 in figure 11), if
it is supposed that the diamagnetic portion cannot hold any
magnetic flux (Ravinder Reddy et al 1995). Then we find
that for the cylinder
√ χr (T2 ) = (1 − fs )/4 and for the slab
χr (T2 ) = (1 − 1/ 2)(1 − fs ), and fs can be found:
√
2[1 − χr (T2 ) − 1]
fs =
fS = 1 − 4χr (T2 )
(4.5.8)
√
2−1
for a cylinder (radius R) and a slab (thickness 2R)
respectively.
Ghivelder et al (1994) studied the
susceptibility of a cylindrical sample consisting of YBaCuO
grains in an Ag matrix and explained the observed
behaviour with the help of the effective-medium theory
(McLachlan et al 1990), which for the incompletely
penetrated intergrain volume (i.e. low AC fields) yields
χ 0 = −1 +
2 1
Ba
Ba
−
(1 − fs )1.5 (4.5.9a)
µ0 jc R
3 µ0 jc R
2 Ba
4Ba
(1 − fs )1.5
−2
.
χ =
µ0 jc R
µ0 jc R
3π
00
(4.5.9b)
A difference in the estimation of fs using different
formulae (4.5.7) and (4.5.8) is found when for example the
model of jmc independent of the magnetic field is no longer
valid. Another reason for the discrepancy could be that the
diamagnetic fraction of the sample volume has changed its
value owing to the temperature dependence of λL or Bc1 .
Characterization by AC susceptibility
So far we have supposed that the grains are perfectly
shielded. Thus, the diamagnetic fraction of the sample
volume, fs , was identical with the volume fraction occupied
by the superconducting grains, vg . This condition requires
the grain dimension Rg to be much larger than the London
penetration depth, λL . Otherwise we must account for the
effect of so-called magnetic invisibility (Clem and Kogan
1987) mentioned in section 3.4. Then, the fraction of
the sample volume screened by the intragrain Meissner
currents, fs , will differ from the volume fraction occupied
by the superconducting grains, vg (Chen et al 1990).
Replacing again the individual grains by a system of
identical cylinders (radius Rg ) and taking into account the
formula (3.4.2) we find that for the fields under Bc1
vg =
1−
fs
2λL I1 (Rg /λL )
Rg I0 (Rg /λL )
.
(4.5.10)
For large grains (Rg λL ) the second term in the
denominator goes to zero. Then vg = fs and the content of
superconducting phase, given by vg , can be determined in
a straightforward way. Otherwise, vg is larger than fs .
In the limiting case of tiny grains with Rg λL , the
denominator of equation (4.5.10) goes to zero and one will
observe a negligible diamagnetic fraction of the sample,
fs , also in the case of a dense, single-phase sample with
vg → 1. Therefore the knowledge of λL as well as the
grain shape and dimension is important (Yang et al 1994a).
In particular for Bi-2223 superconductors with plate-like
grains, the susceptibility depends strongly on the field
orientation (Berling et al 1996a) and this difference can
be used to estimate the degree of texturing (Gömöry et al
1996). For the grains supposed to have the form of slab-like
columns instead of cylinders the formula (4.5.10) should be
modified using tanh(Rg /λL ) in place of the Bessel functions
(Angurel et al 1994).
The previous considerations apply well to dense
samples with the grains oriented parallel to the external
field. In the opposite case (isolated grains in the form of
plates perpendicular to the field) the magnetic moments of
individual grains increase the strength of the effective field,
leading to overestimation of vg .
The existence of two peaks does not necessarily signify
sample granularity. For example in untwinned YBaCuO
single crystals this was explained by a first-order melting
transition in the flux line lattice (Giapintzakis et al 1994,
Billon et al 1996, Tatara et al 1996).
4.6. London’s (λL ) and Campbell’s (λC ) penetration
depths
Determination of the reversible screening depth λ by
the AC magnetization technique requires the preparation
of particular samples that contain grains comparable
in dimensions with expected values of λ (Sargánková
et al 1996), preferably in an insulating matrix (Guerrin
et al 1994).
Small AC amplitudes should be used
to make this effect dominant (Shoenberg 1937). By
fitting the experimental temperature-dependent data to
the theoretical expressions (3.4.1) and (3.4.2) one can
find the appropriate λL (T ) dependence and compare
it with predictions of various microscopic theories of
superconductivity (Mühlschlegel 1974, Riseman et al 1994,
Guerrin et al 1995). Because of the intrinsic anisotropy
of HTSCs, grains should be aligned, e.g. magnetically
(Boolchand et al 1992, Porch et al 1993), and the values of
the penetration depth in two principal directions determined
separately (Athanassopoulou et al 1994).
The first attempt (Takács and Campbell 1988) to
perform the electromagnetic calculations for the case when
the bulk pinning (characterized by the critical current
density jc ) is combined with the reversible motion of
the flux lines (characterized by the Campbell penetration
depth λC ) was followed by Matsushita et al (1991). They
obtained for the slab with thickness 2R the numerical
solutions and approximated then as
1
1 + 3(λC /R)2 + y
(4.6.1a)
y
2
π 3[1 + 2(λC /R)2 ]2 + y 2
(4.6.1b)
χ0 =
χ 00 =
where y = Ba /µ0 jc R. The height of the imaginary peak,
00
, is depressed significantly (Campbell et al 1991) and
χmax
allows λC to be estimated regardless of the pinning strength:
λC =
R
.
√
00
(2π χmax
3)1/2
(4.6.2)
4.7. Irreversibility line
We define the irreversibility line (IL) as the line of zero
critical current density: jC (Birr ) = 0 (Matsushita 1993,
Dubois et al 1996).
Remember that in the formulae for the determination
of jc (see section 4.3), the critical current density is always
proportional to the AC field amplitude Ba . Thus, the
direct determination of the jc = 0 line would require the
measurement at Ba = 0, which is impossible. We can
extrapolate to this line by using the data taken at low
amplitudes or finding the point where the upper part of the
χ 00 peak goes to zero (Dhingra and Das 1993, Noetzel et al
1996). Another possibility is to select a certain level of
critical current density as that approaching closely enough,
from the point of view of the studied problem, the zero
level. Then, instead of determining the hardly accessible
true IL, we can study a ‘practical’ IL defined by jc = jt
(Sun et al 1994a, Mosbah et al 1994) where jt is commonly
chosen in the interval 1–100 A cm−2 (Gömöry and Takács
1993, Fabbricatore et al 1993b, Gömöry et al 1993b). ILs
determined from the susceptibility peaks are exactly the
practical ILs (Loegel et al 1993). This is because, for
the conditions of the χ 00 peak found at different DC fields
superimposed on the AC field with constant amplitude,
the critical current density is always the same, given by
equations (4.3.1). A quantitative comparison of the ILs
determined separately from the AC susceptibility and the
DC magnetization is also possible when we compare the
same extent of irreversibility expressed by identical values
of jt (Gömöry et al 1994).
537
F Gömöry
The difficulty in employing a dynamic AC method
in the determination of a static critical current density
clearly increased when ILs were revealed to be frequency
dependent (Nikolo and Goldfarb 1989, Samarappuli et al
1992). The flux density gradient created in the sample
during the change of the external magnetic field can no
longer be assumed to persist in time, especially in granular
samples (Nikolo and Missey 1994, Pérez et al 1994).
Its relaxation towards equilibrium is more pronounced at
lower frequencies because of the longer cycle duration.
The macroscopic current density determined for example
from the AC susceptibility peak in that case decreases with
decreasing frequency. When the frequency is lowered, the
ILs shift towards lower temperatures because the gain in
the unrelaxed value of current density must compensate for
its loss due to relaxation.
The problem of handling strongly relaxing flux density
gradients was discussed in section 3.3. The approach used
there can be judged as a modification of the original model
of Bean (1962): we assume that the actual current density j
flowing in the sample slightly differs from jc and depends
on the electrical field (Yeshurun and Malozemoff 1988).
In the general treatment of the flux dynamics in section 3
we used for the j (E) dependence a phenomenological
model (3.2). This model together with the electrical field
estimation (3.3.6) is quite sufficient to predict the shift
of the IL with frequency. Thorough calculations of the
electrical fields can be found in the work of Brandt (1995).
The shift of the ILs when the frequency is changed is
illustrated in figure 14. Experimental ILs are determined
from the AC susceptibility peak and from the quasistatic
magnetization loops recorded on a SQUID susceptometer.
Theoretical lines are calculated using the hypothetical j (E)
characteristics of the form (3.2). The only parameter that
had to be fitted was the exponent n (Gömöry et al 1995).
4.8. Pinning potential U0
Another interesting parameter that can be found from the
AC susceptibility data taken at different frequencies is the
height of the barrier that the moving element of the flux line
lattice must overcome when hopping to the next pinning
site (Savvides et al 1992, Kusevic et al 1994, Fábrega
et al 1994, Devos et al 1994, Sun et al 1995a, Mehdaoui
et al 1995). Generally, the flux line can be considered as
a particle moving in a force field created by the potential
wells at pinning sites. The effective depth, Ueff , of the
pinning well in the presence of a flux density gradient can
be separated into two terms, i.e.
Ueff = U0 (T , B)f [j/jc (T , B)]
(4.8.1)
the first of them being a material parameter while the
second reflects the modification of the barrier height by
the Lorentz force. For the models of Anderson and Kim
(1964), Feigelman et al (1989), Zeldov et al (1990) and
Fisher et al (1991) one can find U0 from the slope of the
j (E) curve at j = jc (Welch et al 1990, Cesnak et al
1996):
∂Ueff ∂ ln E =
k
T
.
(4.8.2)
U0 = −jc
B
∂j j =jc
∂ ln j j =jc
538
Figure 14. The irreversibility lines measured at frequencies
of 0.1 (F), 2, 12.3, 123, 1230, 12 300 ( ) Hz on a YBaCuO
melt-grown sample, compared with the lines calculated on
the basis of the thermally activated flux creep model.
•
As we can see from the last expression, the determination
of U0 is affected by the definition of jc except the case
when ∂ ln E/∂ ln j = constant. Exactly this happens for
the logarithmically shaped potential Ueff = U0 ln(jc /j )
analysed by Zeldov et al (1990) and leading to the j (E)
curve of the type j ∼ E n where n = ∂ ln E/∂ ln j .
We can collect the susceptibility data at different AC
field amplitudes much larger than Bp = µ0 j R and process
them by supposing that the thermal activation is only a
small perturbation of the pinning-controlled behaviour (Shi
et al 1995). Then the critical-state model formulae (3.2.8)
or (3.2.9) can be still used but we must replace jc in
these formulae by the electrical field dependent j from
equation (3.2). Including the expression (3.3.5) estimating
the electrical field, we can construct the E(j ) curve and
determine U0 with the help of relation (4.8.2): see figure 15.
There is still a need for discussion if the pinning
potential found in the creep regime should be identical with
the activation energy derived from the Arrhenius plots of
the resistivity in the transition region:
Ua
.
(4.8.3)
ρ = ρ0 exp
kB T
The logarithmic U (j ) dependence found for melt-textured
YBCO by Quin et al (1996) seems to support this idea.
5. Conclusions
The goal of this review was to get together the basic
information necessary for an experimentalist to extract
from the AC susceptibility data of an HTSC sample as
much information as possible. In addition to the complex
Characterization by AC susceptibility
of various models. I would like to mention at least
some of them: L Cesnak, P Fabbricatore, M Forsthuber,
K Fröhlich, G Hilscher, T Holubar, P Lobotka, I Madera,
V Plecháček, I Pochaba, M Polichetti, A Rosová, P Kováč,
S Takács, Yu P Tarenkov. The financial support from the
Slovak Grant Agency under Contracts GAV 2/1087/94 and
GAV 108/96 is also acknowledged.
References
Figure 15. Construction of the E (j ) curve (insert) from the
wide-band AC susceptibility data measured at 85 K, 0.1 T
on a YBaCuO thin film. The portion of the data used for
this purpose is indicated by the arrows. From the
logarithmic slope of the E (j ) curve, the activation energy
was found as U0 /kB = nT = 1445 K.
AC susceptibility, the wide-band AC susceptibilities were
analysed.
The formulae for susceptibilities corresponding to the
models of linear diffusion, reversible screening, critical
state, surface barrier pinning and flux creep can be used
in two ways: for the recognition of the mechanism
controlling the flux dynamics and in the determination of
material properties. We have seen that such important
parameters as the critical temperature, critical magnetic
field, linear resistivity, critical current density, surface
barrier, granularity, volume filling by superconducting
grains, London and Campbell penetration depths, and
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circumstances from the AC magnetization data.
To discuss clearly the principal physical features, the
models used in this paper are the simplest possible.
Therefore I did not present here for example the results
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corresponding sections), or a discussion about the influence
of the sample shape (Chen and Goldfarb 1989, Sun J Z et al
1991, Forsthuber and Hilscher 1992, Clem and Sanchez
1994, Brandt 1994a, 1994b).
Acknowledgments
What could not be referred to in the text of this review is the
support of many colleagues who helped me substantially
either in constructing the apparatus and performing the
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